Calculate combined speed

Question

I have a robotic arm that has to move at a defined speed. it has a speed for the X axis, and it has a speed for the Y axis. So if it moves horizontally it uses the X-axis speed, and if it moves Vertically it uses the Y-axis speed. If it moves diagonally it should combine the two speeds. In the image you can see some samples. the vertical and horizontal speeds are given, the angle between the X axis and the diagonal Line is given too.

What I would need to know is how I can calculate the length of the diagonal line (composite speed) with the given informations.

EDIT: the combined speed should follow the ellypse like in the image.

Answer 1

If the absolute value of velocity has to follow the ellipse its square can be computed as $$v^2=\frac {v_x^2v_y^2}{v_x^2\sin^2\theta+v_y^2\cos^2\theta},$$ where $\theta $ is angle with respect to the $x $-axis.

Answer 2

If you have $v_x$ and $v_y$, use the Pythagorean Theorem to get the total velocity $\sqrt{v_x^2+v_y^2}$.